The category \(\mathbf {Gr}^{\bullet }\)  of groups with action is defined in [1]} as a category, where objects are groups which act on itself. Morphisms in this category are group homomorphisms, which preserve an action operation, i.e. \(\alpha \colon G\rightarrow G^{\prime }\)  is a group homomorphism with the property \(\alpha ({g}^{h})={\alpha (g)}^{\alpha (h)}\) , where \(G,G^{\prime }\in \mathbf {Gr}^{\bullet }\)  and \(g,h\in G\) . The group operation is denoted additively, but it is not commutative in general. Recall that an action \(G^{\prime }\times G\rightarrow G\)  in the category \(\mathbf {Gr}\)  of groups satisfies the following conditions
\({(g+g^{\prime })}^{h} & = {g}^{h}+{g^{\prime }}^{h} \\{g}^{h+h^{\prime }} & = {\left({g}^{h}\right)}^{h^{\prime }} \\{g}^{0} & = g\) 
